Optimal shape design for a time-dependent Brinkman flow using asymptotic analysis techniques

Abstract

In this paper, we consider the geometric inverse problem of recovering an obstacle ω immersed in a bounded fluid flow Ω governed by the time-dependent Brinkman model. We reformulate the inverse problem into an optimization problem using a least squares functional. We prove the existence of an optimal solution to the optimization problem. Then, we perform the asymptotic expansion of the cost function in a simple way using a penalty method. An important advantage of this method is that it avoids the truncation method used in the literature. To reconstruct the obstacle, we propose a fast algorithm based on the topological derivative. Finally, we present some numerical experiments in two- and three-dimensional cases showing the efficiency of the proposed method.

Keywords

Optimal shape design inverse problem time-dependent Brinkman equation topological sensitivity analysis

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References

A.B.

Abda,

Hassine,

Jaoua and

Masmoudi, Topological sensitivity analysis for the location of small cavities in Stokes flow, SIAM journal on control and optimization 48(5) (2010), 2871–2900. doi:10.1137/070704332.

M.S.

Alnæs,

Logg,

K.-A.

Mardal,

Skavhaug and

H.P.

Langtangen, Unified framework for finite element assembly, International Journal of Computational Science and Engineering 4(4) (2009), 231–244. doi:10.1504/IJCSE.2009.029160.

M.S.

Alnæs,

Logg,

K.B.

Ølgaard,

M.E.

Rognes and

G.N.

Wells, Unified form language: A domain-specific language for weak formulations of partial differential equations, ACM Transactions on Mathematical Software 40(2) (2014).

C.J.

Alves and

A.L.

Silvestre, On the determination of point-forces on a Stokes system, Mathematics and computers in Simulation 66(4–5) (2004), 385–397. doi:10.1016/j.matcom.2004.02.007.

Ammari,

Bretin,

Garnier,

Jing,

Kang and

Wahab, Localization, stability, and resolution of topological derivative based imaging functionals in elasticity, SIAM Journal on Imaging Sciences 6(4) (2013), 2174–2212. doi:10.1137/120899303.

Ammari,

Garnier,

Jugnon and

Kang, Stability and resolution analysis for a topological derivative based imaging functional, SIAM Journal on Control and Optimization 50(1) (2012), 48–76. doi:10.1137/100812501.

Angot,

C.-H.

Bruneau and

Fabrie, A penalization method to take into account obstacles in incompressible viscous flows, Numerische Mathematik 81(4) (1999), 497–520. doi:10.1007/s002110050401.

Bergounioux, Introduction au traitement mathématique des images – méthodes déterministes, Vol. 76, Springer, 2015.

Borrvall and

Petersson, Topology optimization of fluids in Stokes flow, International journal for numerical methods in fluids 41(1) (2003), 77–107. doi:10.1002/fld.426.

10.

Carbou,

Fabrie et al., Boundary layer for a penalization method for viscous incompressible flow, Advances in Differential equations 8(12) (2003), 1453–1480.

11.

Caubet,

Conca and

Godoy, On the detection of several obstacles in 2d Stokes flow: Topological sensitivity and combination with shape derivatives, Inverse Problems & Imaging 10(2) (2016), 327. doi:10.3934/ipi.2016003.

12.

Caubet and

Dambrine, Localization of small obstacles in Stokes flow, Inverse Problems 28(10) (2012), 105007. doi:10.1088/0266-5611/28/10/105007.

13.

A.J.

Chorin, Numerical solution of the Navier–Stokes equations, Mathematics of computation 22(104) (1968), 745–762. doi:10.1090/S0025-5718-1968-0242392-2.

14.

Dautray and

Lions, Functional and Variational Methods, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2, 1988.

15.

M.C.

Delfour and

J.-P.

Zolésio, Shapes and Geometries, 2nd edn, Advances in Design and Control, Vol. 22, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011, Metrics, analysis, differential calculus, and optimization.

16.

H.A.

Eschenauer,

V.V.

Kobelev and

Schumacher, Bubble method for topology and shape optimization of structures, Structural optimization 8(1) (1994), 42–51. doi:10.1007/BF01742933.

17.

Evans, Partial Differential Equations, American Mathematical Society. American Mathematical Society, 2010.

18.

J.-L.

Guermond,

Minev and

Shen, An overview of projection methods for incompressible flows, Computer methods in applied mechanics and engineering 195(44–47) (2006), 6011–6045. doi:10.1016/j.cma.2005.10.010.

19.

J.K.

Guest and

J.H.

Prévost, Topology optimization of creeping fluid flows using a Darcy–Stokes finite element, International Journal for Numerical Methods in Engineering 66(3) (2006), 461–484. doi:10.1002/nme.1560.

20.

Guillaume and

K.S.

Idris, The topological asymptotic expansion for the Dirichlet problem, SIAM journal on Control and Optimization 41(4) (2002), 1042–1072. doi:10.1137/S0363012901384193.

21.

Guillaume and

K.S.

Idris, Topological sensitivity and shape optimization for the Stokes equations, SIAM Journal on Control and Optimization 43(1) (2004), 1–31. doi:10.1137/S0363012902411210.

22.

Hassine,

Jan and

Masmoudi, From differential calculus to 0–1 topological optimization, SIAM journal on control and optimization 45(6) (2007), 1965–1987. doi:10.1137/050631720.

23.

Hassine and

Masmoudi, The topological asymptotic expansion for the quasi-Stokes problem, in: ESAIM: Control, Optimisation and Calculus of Variations, Vol. 10, 2004, pp. 478–504.

24.

Henrot and

Pierre, Variation et optimisation de formes, Mathématiques & Applications (Berlin) [Mathematics & Applications], Vol. 48, Springer, Berlin, 2005, Une analyse géométrique. [A geometric analysis].

25.

Lechleiter and

Rienmüller, Factorization method for the inverse Stokes problem, Inverse Problems & Imaging 7(4) (2013), 1271. doi:10.3934/ipi.2013.7.1271.

26.

Logg,

K.-A.

Mardal and

Wells, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Vol. 84, Springer Science & Business Media, 2012.

27.

Mastoid,

Pommier and

Samet, The topological asymptotic expansion for the Maxwell equations and some applications, Inverse Problems 21(2) (2005), 547. doi:10.1088/0266-5611/21/2/008.

28.

Meftahi, Optimal shape design in three-dimensional Brinkman flow using asymptotic analysis techniques, Quarterly of Applied Mathematics 75(3) (2017), 525–537. doi:10.1090/qam/1464.

29.

Pommier and

Samet, The topological asymptotic for the Helmholtz equation with Dirichlet condition on the boundary of an arbitrarily shaped hole, SIAM Journal on Control and Optimization 43(3) (2004), 899–921. doi:10.1137/S036301290241616X.

30.

Samet,

Amstutz and

Masmoudi, The topological asymptotic for the Helmholtz equation, SIAM Journal on Control and Optimization 42(5) (2003), 1523–1544. doi:10.1137/S0363012902406801.

31.

Schumacher, Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungskriterien, PhD thesis, Forschungszentrum für Multidisziplinäre Analysen und Angewandte, 1996.

32.

Sokolowski and

Zochowski, Topological derivative in shape optimization, in: Encyclopedia of Optimization, p. 3908–3918, 2009. doi:10.1007/978-0-387-74759-0_682.

33.

Sturm, Topological sensitivities via a Lagrangian approach for semilinear problems, Nonlinearity 33(9) (2020), 4310. doi:10.1088/1361-6544/ab86cb.

34.

Temam, Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires (ii), Archive for rational mechanics and analysis 33(5) (1969), 377–385. doi:10.1007/BF00247696.

35.

Yan,

Liu and

Jing, Shape inverse problem for Stokes–Brinkman equations, Applied Mathematics Letters 88 (2019), 222–229. doi:10.1016/j.aml.2018.09.003.